/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 59 Solve each equation. $$ 2.4+... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 59

Solve each equation. $$ 2.4+0.4 x=0.25(6+x) $$

Short Answer

Expert verified
The solution is \( x = -6 \).

Step by step solution

01

- Expand the parentheses

First, expand the right side of the equation by distributing 0.25. \[0.25 \times 6 + 0.25 \times x = 1.5 + 0.25x \] So the equation becomes: \[ 2.4 + 0.4x = 1.5 + 0.25x \]
02

- Move like terms together

Subtract 0.25x from both sides to get like terms together. \[ 2.4 + 0.4x - 0.25x = 1.5 \] This simplifies to: \[ 2.4 + 0.15x = 1.5 \]
03

- Isolate the variable term

Subtract 2.4 from both sides to isolate the term with the variable. \[ 2.4 - 2.4 + 0.15x = 1.5 - 2.4 \] This simplifies to: \[ 0.15x = -0.9 \]
04

- Solve for the variable

Divide both sides by 0.15 to solve for \(x\). \[ x = \frac{-0.9}{0.15} \] This simplifies to: \[ x = -6 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

distributive property
When you solve linear equations, the distributive property often helps simplify the equation. The distributive property states that you multiply each term inside a parenthesis by the factor outside. For example, in the equation \( 2.4 + 0.4x = 0.25(6 + x) \), you need to distribute 0.25 to both 6 and \(x\).
This results in:
\( 0.25 \times 6 + 0.25 \times x = 1.5 + 0.25x \)
Remember, always distribute to every term inside the parenthesis. By doing so, the equation becomes easier to solve.
like terms
Combining like terms is a crucial step in simplifying equations. Like terms are terms that have the same variable raised to the same power. In our example, after using the distributive property, the equation becomes:
\( 2.4 + 0.4x = 1.5 + 0.25x \)
To solve the equation, we need to get all the terms involving \(x\) on one side and the constants on the other. Subtract \(0.25x\) from both sides:
\( 2.4 + 0.4x - 0.25x = 1.5 \)
Combine the like terms involving \(x\):
\( 2.4 + 0.15x = 1.5 \).
This step makes it easier to isolate and solve for the variable.
isolating variables
Isolating the variable is essential to solving equations. This means getting the variable term by itself on one side of the equation. Continuing from the combined like terms, our equation is:
\( 2.4 + 0.15x = 1.5 \).
We need to isolate the term with \(x\), so subtract 2.4 from both sides:
\( 2.4 - 2.4 + 0.15x = 1.5 - 2.4 \)
This simplifies to:
\( 0.15x = -0.9 \).
Now, the variable term is isolated, making it easier to solve for \(x\).
solving for variables
The final step is solving for the variable. Once the variable term is isolated, you can easily solve for \(x\). Our equation after isolating is:
\( 0.15x = -0.9 \).
To solve for \(x\), divide both sides by 0.15:
\( x = \frac{-0.9}{0.15} \).
This simplifies to:
\( x = -6 \).
Always check your solution by substituting \(x\) back into the original equation to confirm it satisfies the equation. Practice helps in mastering these steps!

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